A dispensing structure is provided for a container that has an opening to the container interior. A membrane initially occludes the container opening. A cover extends over the container opening and over the membrane. The cover has a peripheral frame and a panel that is connected to the frame. The panel has a dome-like, outwardly convex configuration defining a dispensing aperture and is normally biased outwardly. The panel can be moved to an inverted, inwardly concave configuration. A dispensing valve is disposed in the cover across the dispensing aperture. A penetrator extends from the panel inwardly of the dispensing valve for penetrating the membrane when the panel is in the inwardly concave configuration.